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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 4. Complex Integration}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item {\color{red}Fundamental Theorems}
\begin{enumerate}
\item[1.1.] Line Integrals
\item[1.2.] Rectifiable Arcs
\item[1.3.] Line Integrals as Functions of Arcs
\item[1.4.] Cauchy's Theorem for a Rectangle
\item[1.5.] Cauchy's Theorem in a Disk
\end{enumerate}
 
\item {\color{red}Cauchy's Integral Formula}
\begin{enumerate}
\item[2.1.] The Index of a Point with Respect to a Closed Curve
\item[2.2.] The Integral Formula
\item[2.3.] Higher Derivatives
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3-4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[3.] {\color{red}Local Properties of Analytical Functions}
\begin{enumerate}
\item[3.1.] Removable Singularities. Taylor's Theorem
\item[3.2.] Zeros and Poles
\item[3.3.] The Local Mapping
\item[3.4.] The Maximum Principle
\end{enumerate}

\item[4.] {\color{red}The General Form of Cauchy's Theorem}
\begin{enumerate}
\item[4.1.] Chains and Cycles
\item[4.2.] Simple Connectivity
\item[4.3.] Homology
\item[4.4.] The General Statement of Cauchy's Theorem
\item[4.5.] Proof of Cauchy's Theorem
\item[4.6.] Locally Exact Differentials
\item[4.7.] Multiply Connected Regions
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 5-6}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] {\color{red}The Calculus of Residues}
\begin{enumerate}
\item[5.1.] The Residue Theorem
\item[5.2.] The Argument Principle
\item[5.3.] Evaluation of Definite Integrals
\end{enumerate}

\item[6.] {\color{red}Harmonic Functions}
\begin{enumerate}
\item[6.1.] Definition and Basic Properties
\item[6.2.] The Mean-value Property
\item[6.3.] Poisson's Formula
\item[6.4.] Schwarz's Theorem
\item[6.5.] The Reflection Principle
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{1.1. Line Integrals. Defintion. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}\itemsep0.5em
\item  {\color{red}Question. What is the definition of complex line integral?}

\item  Answer. 
\begin{enumerate}\itemsep0.5em
\item  We consider now a piecewise differentiable arc $\gamma$ with the equation 
$$z = z(t),\, a\le t\le b. $$

\item  If the function $f(z)$ is defined and continuous on $\gamma$, then $f(z(t))$ is also continuous and we can set 
$$
\int_\gamma f(z)dz = \int_a^b f(z(t))z'(t)dt.
$$

\item  This is our definition of the complex line integral of $f(z)$ extended over the arc $\gamma$. 
 
%\item 

%\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Line Integrals. Defintion. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. What is complex line integral with respect to arc length? }

\item  Answer. 
\begin{enumerate}
\item 
An essentially different line integral is obtained by integration with respect to arc length. Two notations are in common use, and the definition is
$$
\int_\gamma fds = \int_\gamma f|dz| = \int_\gamma f(z(t))|z'(t)|dt. 
$$

\item 
This integral is again independent of the choice of parameter. 
In contrast to 
$$
\int_{-\gamma}fdz = \int_\gamma fdz, 
$$
we have now
$$
\int_{-\gamma}f|dz| = \int_\gamma f|dz|. 
$$


\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. Rectifiable Arcs. Defintion. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. What is a rectifiable arc? }

\item  Answer. 
\begin{enumerate}

\item 
The length of an arc can also be defined as the least upper bound of all sums
$$
|z(t_1)-z(t_0)| + |z(t_2)-z(t_1)| + \cdots + |z(t_n)-z(t_{n-1})|
$$
where $a = t_0 < t_1 < \cdots < t_n = b$. 

\item 
If this least upper bound is finite we say that the arc is {\color{blue}rectifiable}. 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. Rectifiable Arcs. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
An arc $z = z(t)$ is rectifiable if and only if the real and imaginary parts of $z(t)$ are of bounded variation.
}

\item  Answer. 
\begin{enumerate}

\item Let $a=t_0 < t_1 <\cdots < t_n=b$ be a partition of the interval $[a,b]$. 

\item The variation of $x(t), a\le t\le b$ is defined to be the least upper bound  of the sums 
$$
|x(t_1)-x(t_0)| + \cdots + |x(t_n)-x(t_{n-1})|. 
$$ 
 
\item The variation of $y(t), a\le t\le b$ is defined to be the least upper bound  of the sums 
$$
|y(t_1)-y(t_0)| + \cdots + |y(t_n)-y(t_{n-1})|. 
$$ 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as Functions of Arcs. \hfill 作业4A-1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 1.
The line integral $$\int_\gamma p dx + q dy,$$ defined in $\Omega$, depends only
on {\color{blue}the end points} of $\gamma$ if and only if there exists a function $U(x,y)$ in $\Omega$ with the partial derivatives $$\partial U/\partial x = p,\,\, \partial U/\partial y= q. $$
}

\item  Proof. 
\begin{enumerate}
\item  The sufficiency follows by direct computation of the line integral. 
 
\item  To prove the necessity we choose a fixed point $(x_0,y_0)\in\Omega$, join it to $(x,y)$ by a polygon $\gamma$, contained in $\Omega$, whose sides are parallel to the coordinate axes. Then we define the function and compute partial derivatives, 
$$U(x,y)=\int_\gamma pds+qdy. $$ 

%\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 1 \hfill 作业4A-2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_\gamma xdz$$ where $\gamma$ is the directed line segment from $0$ to $1+i$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 2 \hfill 作业4A-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=r} xdz,$$ for the positive sense of the circle, in two ways: first, by use of a parameter, and second, by observing that $x=\frac{1}{2}(z+\bar{z})=\frac{1}{2}\left(z+\frac{r^2}{z}\right)$ on the circle. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=2} \frac{dz}{z^2-1}$$ 
for the positive sense of the circle. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 4 \hfill 作业4A-4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=1} |z-1|\cdot |dz|.$$ 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Suppose that $f(z)$ is analytic on a closed curve $\gamma$ (i.e. $f$ is analytic in a region that contains $\gamma$). Show that $$\int_\gamma \overline{f(z)}f'(z)dz$$ is purely imaginary. (The continuity of $f'(z)$ is taken for granted.) 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 6}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Assume that $f(z)$ is analytic and satisfies the inequality $|f(z)-1|<1$ in a region $\Omega$. Show that $$\int_\gamma \frac{f'(z)}{f(z)}dz=0$$ for every closed curve in $\Omega$. ((The continuity of $f'(z)$ is taken for granted.) 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 7}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $P(z)$ is a polynomial and $C$ denotes the circle $|z-a|=R$, compute the value of $$\int_C P(z)d\bar{z}.$$ Answer: $-2\pi i R^2P'(a)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Line Integrals as ... Exercise - 8}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Describe a set of circumstances under which the formula 
$$\int_\gamma \log z dz =0 $$
is meaningful and true.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Cauchy's Theorem for a Rectangle.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 2.
If the function $f(z)$ is analytic on $R$, then
$$\int_{\partial R} f(z) dz = 0. $$
}

\item  Answer. 
\begin{enumerate}
\item This beautiful proof, which could hardly be simpler, is due to E. Goursat who discovered that the classical hypothesis of a continuous $f'(z)$ is redundant. 

\item At the same time the proof is simpler than the earlier proofs inasmuch as it leans neither on double integration nor on differentiation under the integral sign.

%\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.4. Cauchy's Theorem for a Rectangle.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 3. 
Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ by omitting a finite number of interior points $\zeta_i$. If it is true that 
$$ \lim\limits_{z\to \zeta_i}(z - \zeta_i)f(z) = 0$$
for all $j$, then 
$$ \int_{\partial R} f(z) dz = 0. $$ 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.5. Cauchy's Theorem in a Disk. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 4. 
If $f(z)$ is analytic in an open disk $\Delta$, then
$$ \int_\gamma f(z) dz = 0$$
for every closed curve $\gamma$ in $\Delta$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.5. Cauchy's Theorem in a Disk. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 5. 
Let $f(z)$ be analytic in the region $\Delta'$ obtained by omitting a finite number of points $\zeta_i$ from an open disk $\Delta$. 
If $f(z)$ satisfies the condition $$\lim\limits_{z\to\zeta_i}(z-\zeta_i)f(z) = 0$$ for all $j$, then 
$$ \int_\gamma f(z) dz = 0$$
holds for any closed
curve $\gamma$ in $\Delta'$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Index of a Point. Lemma 1. \hfill 作业4B-1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If the piecewise differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral 
$$\int_\gamma \frac{dz}{z-a}$$
is a multiple of $2\pi i$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Index of a Point. Lemma 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Let $z_1, z_2$ be two points on a closed curve $\gamma$ which does not pass through the origin. Denote the subarc from $z_1$ to $z_2$ in the direction of the curve by $\gamma_1$, and the subarc from $z_2$ to $z_1$ by $\gamma_2$. 
Suppose that $z_1$ lies in the lower half plane and $z_2$ in the upper half plane. 
If $\gamma_1$ does not meet the negative real axis and $\gamma_2$ does not meet the positive real axis, then $n(\gamma,0) = 1$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Index of a Point. Exercise 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Give an alternate proof of Lemma 1 by dividing $\gamma$ into a finite number of subarcs such that there exists a single-valued branch of $\mathrm{arg}(z-a)$ on each subarc. Pay particular attention to the compactness argument that is needed to prove the existence of such a subdivision.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Index of a Point. Exercise 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
It is possible to define $n(\gamma,a)$ for any continuous closed curve $\gamma$ that does not pass through $a$, whether piecewise differentiable or not. For this purpose $\gamma$ is divided into subarcs $\gamma_1, \cdots, \gamma_n$, each contained in a  disk that does not include $a$. 
Let $\sigma_k$ be the directed line segment from the initial to the terminal point of $\gamma_k$, and set $\sigma = \sigma_1 + \cdots + \sigma_n$.
We define $n(\gamma,a)$ to be the value of $n(\sigma,a)$.
To justify the definition, prove the following:

\begin{enumerate}
\item  the result is independent of the subdivision;
\item  if $\gamma$ is piecewise differentiable the new definition is equivalent to
the old;
\item  the properties (i) and (ii) of the text continue to hold.
\end{enumerate}
}

\item  Answer. 

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Index of a Point. Exercise 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The Jordan curve theorem asserts that every Jordan curve in the plane determines exactly two regions. The notion of winding number leads to a quick proof of one part of the theorem, namely that the complement of a Jordan curve $\gamma$ has at least two components. This will be so if there exists a point a with $n(\gamma,a) \neq 0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. The Integral Formula.  \hfill 作业4B-2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Theorem 6. 
Suppose that $f(z)$ is analytic in an open disk $\Delta$, and let $\gamma$ be a closed curve in $\Delta$. For any point $a$ not on $\gamma$, 
$$
n(\gamma,a)\cdot f(a) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)dz}{z-a},
$$
where $n(\gamma,a)$ is {\color{blue}the index of $a$ with respect to $\gamma$}. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. The Integral Formula. Exercise -1 \hfill 作业4B-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=1} \frac{e^z}{z}\cdot dz.$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. The Integral Formula. Exercise -2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=2} \frac{dz}{z^2+1}$$
by decomposition of the integrand in partial fractions. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. The Integral Formula. Exercise -3 \hfill 作业4B-4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Compute $$\int_{|z|=\rho} \frac{|dz|}{|z-a|^2}$$
under the condition $|a|\neq \rho$. 
Hint: make use of the equations $z\bar{z}=\rho^2$ and 
$$|dz|=-i\rho \frac{dz}{z}. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Lemma 3. \hfill 作业4B-5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Suppose that $\varphi(\zeta)$ is continuous on the arc $\gamma$. 
Then the function 
$$
F_n(z) = \int_\gamma \frac{\varphi(\zeta)d\zeta}{(\zeta-z)^n}
$$
is analytic in each of the regions determined by $\gamma$, and its derivative is 
$F'_n(z) = nF_{n+1}(z)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Exercise - 1 \hfill 作业4B-6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Compute $$\int_{|z|=1} e^zz^{-n}dz, \,\,\, \int_{|z|=2} z^n(1-z)^mdz, \,\,\,\int_{|z|=\rho} |z-a|^{-4}|dz|\,\, (|a|\neq \rho). $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove that a function which is analytic in the whole plane and satisfies an inequality $|f(z)| < |z|^n$ for some $n$ and all sufficiently large $|z|$ 
reduces to a polynomial.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic and $|f(z)|\le M$ for $|z| \le R$, find an upper bound 
for $f^{(n)}(z)|$ in $|z| \le \rho < R$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic for $|z| < 1$ and $|f(z)| \le 1/(1-|z|)$, find the best
estimate of $|f^{(n)}(0)|$ that Cauchy's inequality will yield. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Higher Derivatives. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that the successive derivatives of an analytic function at a point can never satisfy $|f^{(n)}(z)| > n!n^n$. Formulate a sharper theorem of the same kind.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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